## CLASSIFICATION OF NUMBERS WORKSHEET FOR GRADE 11

Natural numbers:

Set of all numbers which is beginning with 1, 2,... is called as Natural numbers

Symbol for natural number  =

Whole numbers:

Set of all numbers which is beginning with 0, 1, 2, 3...is called as whole numbers.

Symbol for natural number  =  W

Integers:

Integers are set of all whole numbers and their opposites. We are using number line to denote integers....-3, -2, -1, 0, 1, 2, 3....

Symbol for integers  =

Rational numbers

In our number system we are representing the rational number in the form of fraction like a/b.

1.5  =  3/2

Symbol for rational number  =  ℚ

Irrational numbers

An irrational number is any real number which cannot be expressed as a simple fraction or rational number. Π, √2 are some examples or irrational numbers.

Symbol for rational number  =  R - ℚ

Real numbers

All numbers including positive integers, negative integers, rational numbers, irrational numbers, are called real numbers.

It is usually denoted as ℝ

Question 1 :

Classify each element of {√7, −1/4 , 0, 3.14, 4, 22/7} as a member of N, Q, R − Q or Z.

Solution :

√7 is irrational number

√7 ∈ R - ℚ

−1/4  is a rational number

−1/4 ∈ R - ℚ

0 is a integer

∈ Z

3.14 is a rational number

3.14 ∈ Q

4 is a integer

∈ Z

22/7 is a rational number.

22/7 ∈ Q

Question 2 :

Prove that √3 is an irrational number.

(Hint: Follow the method that we have used to prove √2  Q.)

Solution :

Let √3 be a rational number

√3  =  p/q

p and q are co primes.

Co prime means the numbers which has common divisor other than 1.

p = √3 q

Taking squares on both sides, we get

p2 = (√3 q)2

p2 = 3 q2

q= p2/3

p is the divisor of 3.

p/3  =  a

p = 3a

taking squares on both sides, we get

p2  =  9a2

Here p2  =  3q2

(3q2)  =  9a2

q2 =  9a2/3

a2  =  q2/3

q is the factor of 3.

This contradicts out assumption. Hence √3 is irrational number.

Question 3 :

Are there two distinct irrational numbers such that their difference is a rational number? Justify.

Solution :

Let the two irrational numbers be 5 + √3  and 5 - √3.

Difference  =  (5 + √3) - (5 - √3)

=  5 + √3 - 5 + √3

=  2√3

Hence two distinct irrational numbers such that their difference is a rational number

Question 4 :

Find two irrational numbers such that their sum is a rational number. Can you find two irrational numbers whose product is a rational number.

Solution :

Let the two irrational numbers be 5 + √3  and 5 - √3.

Sum  =  (5 + √3) + (5 - √3)

=  5 + √3 + 5 - √3

=  10

Product =  (5 + √3)  (5 - √3)

=  52 - √32

=  25 - 3

=  22

Question 5 :

Find a positive number smaller than 1/21000 . Justify.

Solution :

We see

1/21 > 1/22 > 1/23

Hence the positive number less than 1/21000 is 1/21001

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